Contact Topology and Hydrodynamics

Abstract

We draw connections between the field of contact topology and the study of Beltrami fields in hydrodynamics on Riemannian manifolds in dimension three. We demonstrate an equivalence between Reeb fields (vector fields which preserve a transverse nowhere-integrable plane field) up to scaling and rotational Beltrami fields on three-manifolds. Thus, we characterise Beltrami fields in a metric-independant manner. This correspondence yields a hydrodynamical reformulation of the Weinstein Conjecture, whose recent solution by Hofer (in several cases) implies the existence of closed orbits for all $C^\infty$ rotational Beltrami flows on $S^3$. This is the key step for a positive solution to the hydrodynamical Seifert Conjecture: all $C^\omega$ steady state flows of a perfect incompressible fluid on $S^3$ possess closed flowlines. In the case of Euler flows on $T^3$, we give general conditions for closed flowlines derived from the homotopy data of the normal bundle to the flow.